the topic itself might not be something new, but I don't think this particular problem was posted before. What I am hoping for is advice from somebody who might be more experienced than me on how to tackle the problem I got on my hands. In particular I looking for a solution to the following problem:
Find the probability density of $x$, where $x$ is defined as $x=\sum_{j=1}^{N}{\frac{\cos{\phi_j}}{\sqrt{j}}}$, when $\phi_j$ are independent uniformly distributed random variables between $0$ and $2\pi$.
I have gotten so far to say that:
- The probability density of $\cos{\phi_j}$ is given by $\frac{1}{\pi\sqrt{1-x^2}}$ for $-1 < x < 1$.
- Then it should follow that the distribution of $\frac{\cos{\phi_j}}{\sqrt{j}}$ is given by $\cos{\phi_j}$ is given by $\frac{1}{\pi\sqrt{1-\frac{x^2}{j}}}$ for $-\frac{1}{\sqrt{j}} < x < \frac{1}{\sqrt{j}}$.
So the last step would be to do an N-fold convolution of the aforementioned distributions. As I mentioned in the beginning, I am hoping for somebody who might know of a similar problem or a trick to simplify this problem.
Greets and thank you very much, FB.